The Kobayashi–Royden metric on punctured spheres

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}}, {n\geq 3}, as well, and the metric on {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.

An Equivalent Form of Picard’s Theorem and Beyond

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation f2 + g2 = 1 and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

Integral points of bounded degree on affine curves

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

Value Distribution and Linear Operators

Nevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

NEVANLINNA THEORY FOR HOLOMORPHIC CURVES FROM PUNCTURED DISKS INTO SEMI-ABELIAN VARIETIES

The purpose of this paper is twofold. The first is to show a second main theorem with truncated counting function for holomorphic curves from punctured disks into semi-Abelian varieties. The second is to give an alternative proof of a Big Picard's theorem for algebraically non-degenerate holomorphic curves by Nevanlinna theory.